The Reality of e

Date: 7/26/96 at 15:52:31
From: Anonymous
Subject: The Reality of e

Ok, so e can be defined as a limit or as a sum.  Does it exist outside 
of the classroom?  Can I find it in nature?

Date: 7/26/96 at 22:7:23
From: Doctor Paul
Subject: The Reality of e

A couple of things come to mind.  The first is the use of the function 
e^x to model population growth and population decay.  

P(t) = Po e^(-kt)

where P(t) is the population at time t, Po is the population at t = 0 
and k is decay (or growth) constant. It can be used to solve problems 
of this sort:

This same model can be used to compute interest on a bank account, 
compounded continuously.

Another use of the e^x function is in modeling simple harmonic motion 
(using differential equations).  It could model underdamped, 
overdamped, or critically-damped motion.

Yet another use of the e^x function is in the building of catenaries 
(like the arc in St. Louis).  A catenary is the curve a hanging 
flexible wire or chain assumes when supported at its ends and acted 
upon by a uniform gravitational force.  
y(t) = 1/2*(e^t + e^(-t))

Finally, a form of e^(-x^2) is used in Statistics.  The curve:

f(x) = e^(-.5*x^2)
       -----------
        sqrt(2*Pi)

What's so special about this curve?  The area under it is equal to 
one. That's right: the integral of f(x) from -infinity to infinity 
is 1.  This works out really nicely.  The curve is most commonly 
referred to as a bell curve and is often used for curving tests in 
college.

-Doctor Paul,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/